Related papers: A Trace theorem for Martinet--type vector fields

Let $\alpha>0$, $\beta>\alpha$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{\alpha+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes the…

There are presented certain results on extending continuous linear operators defined on spaces of E-valued continuous functions (defined on a compact Hausdorff space X) to linear operators defined on spaces of E-valued measurable functions…

We characterise the trace spaces arising from intersections of weighted, vector-valued Sobolev spaces, where the weights are powers of the distance to the boundary. These weighted function spaces are particularly suitable for treating…

Using uniformization, Cantor type sets can be regarded as boundaries of rooted trees. In this setting, we show that the trace of a first-order Sobolev space on the boundary of a regular rooted tree is exactly a Besov space with an explicit…

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in $\mathsf{RCA}_0$. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path…

The vector potential is a fundamental concept widely applied across various fields. This paper presents an existence theorem of a vector potential for divergence-free functions in $W^{m,p}(\mathbb{R}^N,\mathbb{T})$ with general $m,p,N$.…

We derive the trace and diffeomorphism anomalies of the Schr\"odinger field minimally coupled to the Newton-Cartan background using Fujikawa's path integral approach. This approach in particular enables us to calculate the one-loop…

For a given $1$-Lipschitz map $u\colon\mathbb{R}^n\to\mathbb{R}^m$ we define a partition, up to a set of Lebesgue measure zero, of $\mathbb{R}^n$ into maximal closed convex sets such that restriction of $u$ is an isometry on these sets. We…

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

Let $E$ be an elliptic curve defined over a number field $K$ with fixed non-archimedean absolute value $v$ of split-multiplicative reduction, and let $f$ be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on…

It is shown that any subset $E$ of a plane over a finite field $\F_q$, of cardinality $|E|>q$ determines not less than $\frac{q-1}{2}$ distinct areas of triangles, moreover once can find such triangles sharing a common base. It is also…

We consider the difference $f(-\Delta +V)-f(-\Delta)$ of functions of Schr\"odinger operators in $L^2(\mathbb R^d)$ and provide conditions under which this difference is trace class. We are particularly interested in non-smooth functions…

We establish half-space type results for a class of height-dependent weighted minimal surfaces in $\mathbb{R}^3$, namely critical points of a weighted area functional whose weight depends on the height. When the weight has at most quadratic…

We classify the polynomials $f(x,y) \in \mathbb R[x,y]$ such that given any finite set $A \subset \mathbb R$ if $|A+A|$ is small, then $|f(A,A)|$ is large. In particular, the following bound holds : $|A+A||f(A,A)| \gtrsim |A|^{5/2}.$ The…

This paper investigates the traces of functions belonging to the inhomogeneous Besov spaces B $\xi$ p,q , where $\xi$ is a product of capacities defined as powers of Gibbs measures. We first establish that the traces of functions in B $\xi$…

We introduce the notion of metric semilattice on the metric space and prove the criterion of $\R$-tree as connected geodesic metric space $X$ admitting the partial order, such that $X$ is semilinear metric semilattice. Also we state the…

We construct an analogue of Kontsevich and Vishik's canonical trace for a class of pseudodifferential boundary value problems in Boutet de Monvel's calculus on compact manifolds with boundary. For an operator A in the calculus (of class…

We complete the proof of the Trace Theorem in the quantized calculus for quasi-Fuchsian group which was stated and sketched, but not fully proved, on pp. 322-325 in the book "Noncommutative Geometry" of the first author.

Consider a finite family $\{f_1,\dots,f_\nu\}$ of $C^\infty$ vector fields on a $n$-dimensional ($n\in\mathbb{N}$), smooth manifold $\mathcal{M}$. The celebrated Rashevskii-Chow theorem states that, provided the vector fields…